1. 9.7 Circles and Lengths of Segments
Geometry
9.7 Circles and Lengths of Segments

2. Theorem
When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. X P S Q R 4 3 x 8 6
8 x 3 = 6 x 4

3. Example Another way to solve would be to think of two numbers
that multiply to 36 and add to 15.
b. If AP = 4, PB = 9, and CD = 15, find CP. P B A D C x 4 P 9
15 - x
9 x 4 = x(15 – x)
36 = 15x – x2
x2 – 15x + 36 = 0
(x )(x ) = 0
CP is either 12 or 3.
x – 3 = 0 or x – 12 = 0
If it were given that the figure was
drawn to scale, then CP = 12.
x = 3 or x = 12

4. Theorem
When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. G H F D E
FD x FE = FH x FG
External x Whole Thing = External x Whole Thing

5. Example a. If PQ = 9, QR = 3, and TR = 4, then SR = ____.
b. If PQ = 11, SR = 12, and TR = 5, find PR. R Q T S P 9
3(12) = 4x 12
36 = 4x x
PR = x + 11 = (4) + 11 = 15
9 = x 3 4
RS = 9
x(x + 11) = 5(12) R Q T S P
x2 + 11x = 60
x2 + 11x – 60 = 0 11
x + 11
(x )(x ) = 0 12
x – 4 = 0 or x + 15 = 0 x 5
x = 4 or x = -15
x must be positive

6. Theorem
When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment. B P C A
PB x PC = (PA)2
External x Whole Thing = (Tangent)2

7. Example 25 5 a. If RT = 25 and RS = 5, find QR.
b. IF QR = 6 and ST = 9, find RS. x
x2 = 5(25)
x2 = 125
x = 5√5
QR = 5√5
x + 9 9 R Q S T x
x(x + 9) = 62
x2 + 9x = 36
x must be positive.
x2 + 9x – 36 = 0 6
(x )(x ) = 0
x – 3 = 0 or x + 12 = 0
x = 3 or x = -12
RS = 3

8. If RQ=8, QS = 6, and TQ = 16, then QU = ____.
Complete.
If RQ=8, QS = 6, and TQ = 16, then QU = ____.
2. If RS =13, RQ = 7, and QU = 3, then TQ = ____.
3. If TU = 10√2, QU = 2√2 , and RQ = 8, then RS ____.
4. If RS = 17, TQ = 18, and QU = 4, then RQ = ___. 3 Q T R U S 14 Q 12
8 or 9
Who can fill in the picture on the front board?

9. 5. If DF = 17, EF = 3, and GF = 6, then HF = ____.
Complete.
5. If DF = 17, EF = 3, and GF = 6, then HF = ____.
6. If HG = 8, GF = 7, and DF = 21, then EF = ____.
7. If GF = 8, HG = 10, and DE = 18, then EF = ____.
8. If DF = 25, EF = 4, and HG = GF = x, then x = ____. F H G E D 8½ 5 6
5√2

10. Secants and tangents are shown. Find the value of x.
9) ) ) ) 16 x 15 5 x 4 x 12 3x x 8
x = 6
x = 3
x = 9
x = 4

11. HW Be nice to someone, take the trash out
P WE(1-9, 13-20, 24)